I was wondering if the unit cell (of a given lattice) had to have every point group symmetries of the lattice it defines ? I guess there is no unique way to define a unit cell and that it may not have all point group symmetries. However, is it possible to define a unit cell that has all point group symmetries?
You can certainly take a unit cell which has the full point group symmetry and distort it in such a way that the symmetry is reduced.
The Wigner-Seitz Cell provides you with a procedure to construct a unit cell which has the full point group symmetry. It is defined as the volume of points which are closer to one lattice point than to any other one. The interesting question would now be, whether or not the Wigner-Seitz Cell is uniquely specified given the point group symmetry and the requirement of the unit cell to be a primitive one. I am not aware of a proof of this statement.